An Internal Spur Gear and Pinion is a gear pair where a small external pinion meshes with the inward-facing teeth of a larger ring gear, both shafts running parallel and rotating in the same direction. You see it inside a Toyota Aisin AW automatic transmission, where the planetary ring gear engages the sun-pinion through this exact geometry. The arrangement transmits power in a compact footprint with shorter centre distance than two external gears of the same ratio. That gives you higher ratios, better tooth contact, and a stiffer drive in the same envelope.
Internal Spur Gear and Pinion Interactive Calculator
Vary module, tooth counts, and pinion speed to see internal center distance, ratio, ring speed, and compactness versus an external pair.
Equation Used
The calculator uses the article geometry for an internal spur gear: the pinion sits inside the ring, so center distance is half the pitch diameter difference, not half the sum. Gear ratio is based on tooth count, and the driven internal ring rotates in the same direction as the pinion.
- Standard spur gear pitch diameters use D = m * N.
- Ring tooth count is greater than pinion tooth count.
- Pinion drives the internal ring gear with both gears rotating the same direction.
- Ideal geometry only; backlash, profile shift, efficiency, and interference checks are not included.
How the Internal Spur Gear and Pinion Actually Works
The Internal Spur Gear and Pinion, also called the Internally toothed spur gear and pinion in older British engineering texts, works by meshing an external pinion with teeth cut on the inside diameter of a larger annular gear. Both shafts run parallel. Both rotate in the same direction → that is the key visual difference from a normal external pair, which counter-rotates. Because the pinion sits inside the ring rather than beside it, the centre distance shrinks to (Dring − dpinion) / 2 instead of (D1 + D2) / 2. You get a more compact drive for the same ratio, and the teeth share contact over a longer arc, so tooth loading per pair drops.
The involute tooth profile on an internal gear is concave instead of convex. That concave-on-convex contact means the relative radii of curvature at the mesh point are closer in magnitude, which lowers Hertzian contact stress at the same torque. In plain terms — the teeth rub less and last longer. But you pay for it in manufacturing. You cannot hob an internal gear the way you hob an external one. You shape it with a pinion-type cutter on a Fellows shaper, or you broach it, or you wire-EDM the profile on small precision rings. Get the cutter geometry wrong and you produce trochoidal interference at the tip — the pinion tooth tip gouges the root fillet of the ring gear and the whole set scraps.
Tolerances matter. The centre-distance tolerance on a planetary ring is typically ±0.025 mm on automotive applications. Push that wider and you get backlash variation around the rotation, which shows up as a low-frequency whine you can hear at idle. Push it tighter without matching the tooth profile and the pair binds when warm because aluminium carriers expand faster than steel rings. The classic failure modes are tip interference (audible click per revolution), root fatigue cracking (catastrophic, usually at 60-70% of rated cycles when overloaded), and pitting on the ring teeth from inadequate lubrication film at the contact point.
Key Components
- Internal ring gear (annulus): The large gear with teeth cut on its inside diameter. Pitch diameter sets the outer ratio. On a typical automotive planetary, the ring gear runs 80-120 mm pitch diameter with 60-80 teeth at module 1.5-2.0. Hardness target is 58-62 HRC after case carburising.
- External spur pinion: The smaller driving (or driven) gear sitting inside the ring, with teeth cut on its outside diameter. Pitch diameter and tooth count set the input side of the ratio. Pinion tooth count usually stays above 17 to avoid undercutting on a standard 20° pressure angle profile.
- Bearing supports: Both shafts must run on rigid bearings — the centre distance is fixed by the housing, and any radial float shows up as backlash and noise. Deep-groove ball bearings are standard, with radial play class C2 or tighter for quiet operation below 60 dBA.
- Lubrication interface: The mesh point sits inside the ring, so splash lubrication often misses it. Most production designs use forced oil jets or grease-pack lifetime fills with NLGI 2 lithium-complex grease. Inadequate film thickness drops contact-fatigue life by an order of magnitude.
- Tooth profile (involute, 20° pressure angle): Internal teeth use a concave involute on the addendum side. This concave-on-convex contact is what gives the internal pair its lower contact stress versus an equivalent external pair carrying the same torque.
Real-World Applications of the Internal Spur Gear and Pinion
The Internal Spur Gear and Pinion shows up wherever engineers need a high reduction in a small package, the same shaft direction on input and output, or a compact rotating ring driven by a small motor. The internally toothed spur gear and pinion is the heart of every planetary gearset, every automotive starter ring, and every slewing-bearing crane drive. You will find it in the ratio range of about 2:1 up to 8:1 per stage, with practical limits set by tooth interference at the low end and pinion undercutting at the high end.
- Automotive transmissions: The ring gear in every Toyota Aisin AW 8-speed planetary automatic, where the sun pinion meshes with the internal ring teeth to deliver overdrive ratios.
- Engine starting systems: The flywheel ring gear on a Bosch starter motor — the Bendix-drive pinion engages the flywheel internal teeth at roughly 250 RPM cranking speed to spin the engine to firing speed.
- Construction equipment: Slewing drives on Liebherr LTM mobile cranes use a large internally toothed ring driven by a pinion off a hydraulic motor to rotate the entire upper structure.
- Wind turbines: The Vestas V90 gearbox uses planetary stages with internal ring gears feeding the high-speed generator shaft, converting roughly 16 RPM rotor input to 1500 RPM generator output.
- Robotics and servo drives: Harmonic Drive and Nabtesco RV cycloidal reducers use internal ring teeth meshed with a pinion or wave generator to deliver 30:1 to 320:1 ratios for industrial robot joints.
- Power tools: Cordless drill planetary gearheads from Makita and DeWalt use a 3-stage internal ring and pinion system to drop motor speeds of 20,000 RPM down to chuck speeds of 400-2000 RPM.
The Formula Behind the Internal Spur Gear and Pinion
The gear ratio for an Internal Spur Gear and Pinion is the ring tooth count divided by the pinion tooth count. Sounds trivial — but the ratio you can practically achieve depends on operating range. At the low end of typical planetary stages (around 2.5:1) the pinion is large relative to the ring and tooth-tip clearance is generous. At the nominal 4-5:1 sweet spot the geometry behaves cleanly with a 17-20 tooth pinion against a 70-100 tooth ring. Push past 8:1 in a single stage and you start running into pinion undercutting or ring-tooth pointing — that's why you stage the reduction instead.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| i | Gear ratio (input speed / output speed when ring is output and pinion is input) | dimensionless | dimensionless |
| Nring | Number of teeth on the internal ring gear | teeth | teeth |
| Npinion | Number of teeth on the external spur pinion | teeth | teeth |
| C | Centre distance between pinion and ring axes | mm | in |
| m | Module (metric tooth size, m = pitch diameter / tooth count) | mm | in (use diametral pitch) |
Worked Example: Internal Spur Gear and Pinion in a coffee-roaster drum drive
Suppose you are sizing the internal ring drive on a 12 kg batch coffee roaster — a Probat-style sample roaster rebuild — where a 0.37 kW gear motor running at 1400 RPM must drive the perforated roasting drum at a target 60 RPM through a single internally toothed ring pressed onto the drum end-cap. The pinion sits on the gear motor output shaft. You have room for a ring gear up to 180 mm pitch diameter inside the drum-bearing housing, and you want to confirm tooth counts, centre distance, and what the drum speed actually does across the gear motor's operating range.
Given
- Motor speed (nominal) = 1400 RPM
- Target drum speed = 60 RPM
- Module m = 1.5 mm
- Pinion tooth count Npinion = 20 teeth
- Maximum ring pitch diameter = 180 mm
Solution
Step 1 — calculate the required ratio at nominal motor speed:
That ratio is too high for a single internal-pinion stage. Single-stage internal sets max out around 8:1 cleanly. So we redesign for a 2-stage drive — a belt or pulley pre-reduction of 3:1 dropping motor input to 467 RPM at the pinion, then a single internal ring stage handles the rest.
Step 2 — find the ring tooth count for the internal stage at nominal:
156 teeth at module 1.5 gives a pitch diameter of 234 mm — too big. Drop to module 1.25 and recompute: pitch diameter = 156 × 1.25 = 195 mm. Still over the 180 mm envelope. Trim ring teeth to 140 (ratio becomes 7:1), accept output speed of 467 / 7 = 66.7 RPM. Pitch diameter = 140 × 1.25 = 175 mm. Fits.
Step 3 — centre distance for the internal pair:
Now check the operating range. At the low end of the gear motor's usable range (say 700 RPM motor speed for a slow first-crack roast), the drum sees 700 / 3 / 7 = 33.3 RPM — that's the speed where bean agitation becomes lazy and roasting becomes uneven. At nominal 1400 RPM motor speed, drum runs at 66.7 RPM — the sweet spot for even Maillard development. Push the motor to 1700 RPM (high end via VFD overspeed), drum hits 81 RPM, and beans start sliding rather than tumbling because centripetal force at the drum wall begins to dominate gravity for 12 mm green beans.
Result
Final design: 20-tooth pinion meshing with a 140-tooth internal ring at module 1. 25, centre distance 75 mm, delivering a nominal 66.7 RPM drum speed from a 1400 RPM motor through a 3:1 belt pre-reduction. That's the sweet spot — beans tumble cleanly without sliding. At the 33 RPM low end (motor at 700 RPM) you'll see lazy agitation and scorching on the drum wall; at the 81 RPM high end beans start centrifuging instead of falling, which kills convection heat transfer. If you measure 60 RPM but read motor current 30% above predicted, suspect (1) ring-gear-to-pinion centre distance machined undersize causing tight mesh and binding when warm, (2) pinion bore runout above 0.05 mm TIR producing once-per-rev torque pulses that show as ammeter flutter, or (3) lubrication starvation at the internal mesh point — splash lube rarely reaches the top of an internal ring without a baffle.
When to Use a Internal Spur Gear and Pinion and When Not To
An Internal Spur Gear and Pinion is one of three common ways to get a high reduction in a small space. The Internally toothed spur gear and pinion competes mainly with external spur pairs and with worm-and-wheel drives. Each wins on different dimensions, and the choice depends on whether you need compactness, efficiency, or self-locking behaviour.
| Property | Internal Spur Gear and Pinion | External Spur Gear Pair | Worm and Wheel |
|---|---|---|---|
| Single-stage ratio range | 2:1 to 8:1 | 1:1 to 6:1 | 5:1 to 100:1 |
| Mesh efficiency | 97-98% | 98-99% | 40-90% (ratio dependent) |
| Centre distance for given ratio | Compact — (D−d)/2 | Wide — (D+d)/2 | Perpendicular axes |
| Manufacturing cost (relative) | High (Fellows shaping or broaching) | Low (hobbing) | Medium (thread grinding) |
| Backlash typical (arc-min) | 3-8 | 5-15 | 10-30 |
| Self-locking capability | No | No | Yes (lead angle <5°) |
| Service life at rated load | 10,000-25,000 hr | 15,000-30,000 hr | 5,000-15,000 hr |
| Best fit application | Planetary gearsets, ring drives, starter rings | General-purpose reductions, parallel shafts | High reduction with shaft turn, hoists |
Frequently Asked Questions About Internal Spur Gear and Pinion
That's almost always ring ovality, not a tooth problem. When you press an internal ring into a thin-wall housing or a drum end-cap, the ring distorts into a slight oval — typically 0.05-0.15 mm out of round. The mesh tightens twice per revolution at the minor axis and loosens at the major axis. You hear a 2× rotational frequency hum, not the much higher tooth-mesh tone.
Diagnostic check — pull the ring and measure across four equally spaced diameters with a bore mic. If max-min exceeds 0.05 mm, machine the housing bore round, or shrink-fit the ring with a controlled interference of 0.02-0.04 mm rather than press-fit.
Single stage wins on cost, parts count, and efficiency — one internal mesh at 97% beats two stages at 0.97 × 0.97 = 94%. Single stage loses on torque density and noise. A 7:1 internal stage carrying the full output torque demands a wider face width and heavier ring than a first stage carrying 1/3 of the output torque.
Rule of thumb — if output torque is below 50 Nm, go single stage. If you're above 200 Nm or you need quiet operation below 65 dBA, split it. Between 50 and 200 Nm, the call goes on whether you have axial space (single stage) or radial space (two stages).
One-flank pitting almost always means unidirectional loading combined with insufficient EHL film thickness on that flank. The internal mesh point sits at the top of the ring rotation in most installations, and gravity drains lubricant away from it between revolutions. The drive flank wets fine on the down-stroke but starves on the up-stroke.
Check the lubricant viscosity at operating temperature — if your kinematic viscosity at the mesh temperature drops below about 50 cSt, you're running mixed-film instead of full-film EHL. Switch to a higher-VI synthetic, or add a windage baffle that throws oil onto the upper mesh point.
Yes — these are the same mechanism with different industry names. "Ring gear and pinion" is the automotive and starter-motor term. "Internally toothed spur gear and pinion" is the formal engineering description used in textbooks and gear-design standards. "Annulus and planet" is the planetary-gearbox term. All three describe a small external pinion meshing with the inside-cut teeth of a larger ring on parallel shafts rotating the same direction.
For a standard 20° pressure angle involute internal mesh, the practical minimum pinion tooth count is around 14-15 teeth, but the safer lower bound is 17. Below 14 you start getting trochoidal interference at the pinion tip cutting into the ring root fillet — the pair will mesh on a bench but seize under load.
You can push to 12 teeth if you switch to a 25° pressure angle and apply positive profile shift on the pinion (typically x = +0.3 to +0.5). That's how power-tool planetaries get away with 11-tooth sun pinions. But for general design, hold 17 minimum and you'll never hit interference.
Differential thermal expansion between an aluminium carrier and a steel ring gear. Aluminium expands at roughly 23 µm/m/°C, steel at 12 µm/m/°C. Over a 60°C rise on a 175 mm pitch diameter, the aluminium housing grows about 0.24 mm in diameter while the steel ring grows 0.13 mm. That 0.11 mm differential opens the centre distance and adds backlash.
Two fixes — match materials (steel housing for steel ring), or design with a controlled thermal preload using a tapered roller bearing that takes up clearance as the housing expands. Most production planetary gearboxes use the second approach because aluminium housings save weight.
References & Further Reading
- Wikipedia contributors. Epicyclic gearing. Wikipedia
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